Radial profiles of the coefficients A (blue), B (coral), and C (purple) from the linear PDE system (Eq. (9)) found by SINDy, governing the evolution of velocity perturbations in small impact simulations (Eq. (3)). The black lines show the corresponding analytical values derived in Sect. 4 (Eq. (11)). The excellent agreement between the SINDy-discovered coefficients and the analytical predictions demonstrates the success in recovering the underlying linear dynamics.